adaptive observer-based controller design for a class of nonlinear

8/12/2019 Adaptive Observer-based Controller Design for a Class of Nonlinear

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Adaptive Observer-based Controller Design for a Class of Nonlinear

Systems with Application to Image Guided Control of Steerable Needles

M. Motaharifar, H.A. Talebi, A. Afshar, and F. Abdollahi

Abstract Flexible needles with a bevel tip (steerable needles)promise to enhance targeting accuracy and maneuver inside thehuman body in order to avoid collision with delicate organs.Contributing image feedback to needle insertion tasks greatlyimproves such objectives. An important issue in 2D motionplanning tasks is stabilizing the needle in a desired plane. Anydivergence from the plane leads to the inefficiency of the motionplanning scheme. Hence, a control scheme is proposed in thispaper which guides the needle to a desired plane. The systemof such task is subject to parametric uncertainty. Although theoriginal system is linearly parametrized, the feedback linearizedform is not, which prevents the application of conventionaladaptive control schemes. Moreover, all state variables of thesystem could not be measured and a nonlinear observer isnecessary to observe the system states. In this paper, thepreviously proposed adaptive state feedback controller for suchsystems is modified to an adaptive output feedback controllerand the proposed scheme is applied to the problem of needleguidance. Simulation results are presented to illustrate theenhanced performance of the proposed controller methodologyas compared to previously proposed feedback linearizationscheme.

I. INTRODUCTION

Needle insertion into soft tissue is a widespread surgical

technique that has numerous applications in medical tasks

such as brachytherapy, anesthesia, biopsy, etc. Accurately

placing the needle is a key factor that determines the effec-

tiveness of the treatment. Moreover, the needle should avoidcollision with some sensitive organs such as nerves, bones

or vessels to prevent subsequent complications.

The traditional rigid needles have a little manoeuvrability

in the tissue. Motion planning for such needles just involve

optimization of initial parameters such as needle starting

insertion point, heading angle, etc [1],[2]. On the other hand,

flexible needles have more manoeuvrability than the rigid

ones and can be manipulated during the insertion to guide

through tissue. A class of flexible needles with a bevel-tip

have attracted great attention during the past several years.

Indeed, the asymmetry of the tip causes the needle to bend

and follow a circular arc with constant curvature[3]. There-

fore, with appropriate rotation of the needle from the base,the desired trajectory could be achieved. A mathematical

nonholonomic needle steering model has been developed

and validated in [4] for such needles. The proposed model

was a generalization of the standard 3 DOF nonholonomic

unicycle and bicycle models to 6 DOF using Lie group

theory. Based on this model, some motion planning algo-

rithms have been proposed for this class of needles using

The authors are with Electrical Engineering Department, Amirkabiruniversity of Technology , Tehran, Iran, email: {md.motaharifar,alit, aafshar, f abdollahi}@aut.ac.ir

Stochastic Motion Roadmap [5], inverse kinematics [6],

screw-based motion planning [7], and Rapidely-exploring

Random Trees(RRT) [8]. In [9], a motion planning approach

was proposed based on fast duty cycle spinning of the needle

in order to remove the limitation of a fixed curvature path.

Since a continuous spinning increases patient trauma, this

method is not appropriate for clinical diagnosis. On the other

hand, manual needle insertion using master-slave devices

is another area of research. In [10], the performance of

manual teleoperation scheme was compared to that of an

automatic needle insertion technique and it was shown that

the hybrid control provided improved accuracy. A review onsome research of needle insertion strategies was given in

[11].

Guiding and stabilizing a flexible needle to a desired plane

is a first step in 2D motion planning. In [12], a feedback

linearization controller was employed to do such task. The

proposed controller was based on the reduced order system of

the nonholonomic needle steering model introduced in [4].

Such model, however, is subject to parametric uncertainty

and the feedback linearization controller cannot tolerate

uncertainties. In [13], a way of combining such controllers

with a previously proposed motion planning algorithm has

been presented. Moreover, this class of controllers have the

ability to combine with manual teleoperated insertions.In this paper, an adaptive output feedback controller is

proposed for guiding flexible needles to a desired plane.

The needle steering model considered here belongs to a

special class of nonlinear systems which the original system

is linearly parametrized but the canonical form is not. The

traditional approaches for adaptive output feedback controller

design (e. g. [14], [15]) cannot be applied to a system

unless its canonical form model is linearly parametrized.

Moreover, the unmeasured states of the system have to be

estimated. Some adaptive state feedback controllers were

proposed for this class of nonlinear systems (e. g. [16],[17]).

However, those adaptive control schemes rely on full state

measurement and to the best of our knowledge no work withadaptive output feedback controller has been proposed for

this class of nonlinear systems. The proposed controller is

a combination of the state feedback controller proposed in

[16] with high gain observers after some modifications. As

will be explained in Section III, some complexities emerge in

case of designing an adaptive output feedback version based

on the approach stated in [16]. First, this method is based on

an auxiliary dynamic equation with some properties which

simplify the design. In case of existing unmeasured state

variables, the auxiliary dynamic does not have the simplifier

2012 American Control Conference

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properties which complicates the design. Moreover, some

complexities occur in dealing with the lyapunov function.

The applied controller has the ability in online estimation of

needle curvature. This parameter depends on the needle and

the tissue. The previous work [4] estimates this parameter

off-line using trial and error strategy with some experiments.

Since inserting the needle into the patients body for so many

times is dangerous for the patients health, this method is not

applicable for clinical activities. In brief, the contribution

of this paper is twofold. First designing an adaptive output

feedback for a class of nonlinear systems which the original

(not canonical form) model is linearly parametrized and

output feedback linearizable. Second, estimation of needle

curvature in medical tasks online.

The rest of this paper is organized as follows. In Section

II, the System dynamics is described. Our proposed adaptive

observer-based controller is presented in Section III. Section

IV depicts simulation results of the proposed methodology.

Finally, the conclusions are stated in Section V.

I I . SYSTEM D ESCRIPTION ANDM ODEL

A flexible bevel-tip needle can be steered by rotation

and insertion at the base outside the body of the patient.

Such a needle bends as it is inserted into tissue at a

constant curvature, which is a property of the needle and

tissue. Hence, by rotating the needle from the base, different

trajectories could be achieved.

A kinematic bicycle model is developed for such a needle

in [4]. Based on this model, a reduced order model is

extracted for stabilizing the needle to a desired plane in [12].

This model is reproduced here for reader convenience.

Fig. 1 shows the kinematic bicycle model. In this model,

frame A is the reference frame and frames B and C areattached to the two wheels of the bicycle. Utilizing Lie-

group theory, a coordinate-free differential kinematic model

is found [4].

v

= u1V1+ u2V2 (1)

where v, R3 denote the linear and angular velocities ofthe needle tip, respectively, written relative to frame A. u1and u2 are the insertion and rotation speed of the needle,

and

V1 = e3

e1 and V2 = 033

e3 (2)

The unit vectors ei, i = 1, 2, 3 are the standard basis. Letq= [x,y,z,,, ]be the position and orientation vector ofthe needle tip where x,y,and z are position relative to the

reference frame, is the yaw of the needle in the plane, is

the pitch of the needle out of the plane, and is the roll of

the needle. Moreover, denotes the curvature which needle

follows. Body frame velocity may be expressed as

v

= Jq (3)

Fig. 1. Kinematic bicycle model [4] used with permission from the authors

where

J=

RTAB 033033 S

S=

coscos sin 0 cossin cos 0

sin 0 1

(4)

where RAB is the rotation matrix between frames A and

B. Now, using (1) and (3) the flexible bevel-tip needle model

is

q= J1V1u1+ J1V2u2 =

sin 0 cossin 0

cos cos 0 cossec 0 sin 0

cos tan 1

u1u2

(5)In order to stabilize the needle to the yz plane, the statesy,z, and need not be controlled. Moreover, these states do

not affect the dynamics of the remaining states. Hence, we

can define pT = [p1, p2, p3] = [x,,] as the state vectorof the reduced order system, which can be represented as

follows.

p= f(p)u1+ g(p)u2

=

sin(p2) sin(p3) cos(p3) tan(p2)

u1+

00

1

u2 (6)

r= h(p) = p1 (7)

Note that pT

=

0 0 0

is the desired equilibrium pointof the system which corresponds with placing the needle in

y z plane. we divide both sides of (8) by u1. Hence, thesystem is reparametrized in terms of insertion distance, l.

Note that wherever we write p, we mean dpdl

. In other words,

the insertion distance is substituted for t as the independent

variable and its derivative u1 is no longer an input signal.

Indeed, The resulted system is [12]

p=

sin(p2) sin(p3) cos(p3) tan(p2)

+

00

1

u (8)

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where u = u2u1

r= h(p) = p1 (9)

In essence, we can only measure p1 = x by imageprocessing and the other state variables should be estimated.

Furthermore, the parameter is the unknown curvature of

the needle which should be estimated.

The system (8) and (9) can be transformed into output

feedback linearized form using the following transformations[19]

w= [h(p), Lfh(p), L2fh(p)]

= [p1, sinp2, cosp2sinp3] (10)

v= L3fh(p) + LgL2fh(p)u

= 2 sinp2+ cosp2cosp3u (11)

where Lfh(p) is the Lie derivative ofh with respect to f,defined by [19]

Lfh(p) = h(p)

p f(p) (12)

Indeed, this is the familiar notation of the derivative of h

along the trajectories of the system p= f(p). Moreover, ina similar manner we have

LgLfh(p) =Lfh(p)

p g(p) (13)

Lkfh(p) =Lk1f h(p)

p f(p) (14)

Now, the transformed system is

w = Aw + Bv =

0 1 00 0 1

0 0 0

w+

00

1

v (15)

r= Cw =

1 0 0w (16)

Obviously, the system (15) and (16) is not linearly

parametrized. Therefore, the traditional Model Reference

Adaptive Control (MRAC) [20] cannot be applied to it.

III . THE P ROPOSEDS CHEME

Consider a nonlinear system, subjected to parametric un-

certainty, described as

x= f(x,u,) = f0(x, u) + fT1 (x, u) (17)

y= h(x, u) (18)

where x Rn is the state vector, u Rm is the control

input vector, and Rp is the parameter vector. Moreover,f0 and f1 are known nonlinear functions. Obviously, the

system (8) and (9) can be stated as (17) and (18). Our

objective is to design an adaptive controller-observer pair

for the above system such that the stability is preserved

and tracking a reference signal xd(t) is achieved in thepresence of unknown parameter vector . In order to do so,

the adaptive controller given in [16] is combined with a

nonlinear high gain observer with some modifications.

Assumption:There exist

(a)a Hurwitz matrix A

(b)an open set Dx Rn containingxd(t) for all t

(c)an open set D Rp containing

(d)a family of parametrized diffeomorphisms

W :Dx Rn :z = W(x,) (19)

exists where is an estimation of and such that thefollowing implicit equation in the unknown u

W

x(x,)[f0(x, u) + f

T1 (x, u)]

= W

x(xd,)xd A[W(xd,) W(x,)] (20)

has a unique bounded solution u= ua(x, xd,) for all x Dx. Now, taking the derivative ofz we have

z= W

x(x,)[f0(x, u) + f

T1 (x, u)] +

W

(x,) (21)

The previous equation is equal to

z= g0(x,u,) + gT1(x,u,

)+ g2(x,) (22)

where

g0(x,u,) = W

x(x,)f0 (23)

gT1(x,u,) = W

x(x,)fT1 (24)

gT2(x,) = W

(x,) (25)

Transformation (19) should convert the system (17) and

(18) to normal form. Indeed,

z = z+ B(x, u)y = Cz

(26)

where

=

0 1 0...

.... . .

...

0 0 10 0 0

BT =

0 0 1

C=

1 0 0

(27)

A high gain observer can be designed for the above system

as [19]z= z+ B0(x, u) +H(y y)y= Cz

(28)

wherezis the estimation ofz. The observer gainHis chosen

as

HT =

1

22

nn

(29)

where is a positive constant to be specified and the positive

constantsi are chosen such that the roots of

sn + 1sn1 + + n1s+ n = 0 (30)

are in the left-half plane. The function 0(x, u) is a nominalmodel of(x, u). The observer error is in the following form

z= z+ B(x, u) HCz (31)

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wherez = zzand (x, u) = (x, u)0(x, u). The aboveequation is equal to

z= Ao,z+ B(x, u) (32)

where

Ao,=

1

1 0

......

. . ....

n1n1 0 1n

n 0 0

(33)

Lemma 1 [18]: For the matrix E= diag(1,,...,n1), wehave the following facts:

(a)

Ao, = 1

E1AoE, B=

1E1B (34)

where

Ao =

1 1 0

......

. . ....

n1 0 1n 0 0

BT =

0 0 n

(35)

(b) Let the positive definite matrix S be the solution of

AToS+ SAo = Q, then

ATo,S+ SAo, = 1ETQE (36)

where S= ETSE

In order to find an adaptation law, the dynamic equation

(22) should be stated in terms of observed states.

z = g0(x,u,)+gT1(x,u,

)+g2(x,)+1(x,x, u) (37)

where the definition ofgi for i = 0, 1, 2 is similar to thosestated in (23) , (24) and (25) and 1(x,x, u) is a bounded

uncertainty. Adaptation law can be stated as follows

= g0(x,u,) + gT1(x,u,

)

+[gT2(x,)g1(x,u,)P ][z ] (38)

= g1(x,u,)P[W(x, ) ] (39)

whereis the auxiliary variable. Moreover, is an arbitraryHurwitz matrix and the positive definite symmetric gain

matrix Pis the solution of the following Lyapunov equation.

TP+ P = Q (40)

where Q is an arbitrary positive definite matrix.

The special statement of (37) and definition (38) will make

the proof of the proposed approach possible. Now, the error

system can be written as

z

=

g

T1(x ,u,) 0

g1(x,u,)P 0 00 0 A0

z

+

1(x,x, u)0

2(x,x, u)

(41)

where =z , =

Assumption: It is assumed that

1 = sup[1(x,x, u)] (42)

2 = sup[2(x,x, u)] (43)

Theorem 1: For the error system (41), , and z arebounded. Moreover, the tracking error is bounded.

Proof: The following Lyapunov function is considered for

the system

V( ,,z) = TP+T+ zTSz (44)

By computing the derivative of (44) and using lemma 1 we

have

V = TQ+ 2TP1(x,x, u) 1zTETQEz

+2

(x,x, u)BT Sz (45)

then we can state

V ||||2(min(Q) 2||P||1||||

)

||z||2(min(Q)

2n2||S||||z|| )

(46)

wherez = Ez. For any positive that ||z|| there existsa positive in the following range

n < min(Q)22||S||

(47)

such that the derivative of the lyapunov function is negative,

provided that

|||| 2 ||P||1

min(Q)

(48)

Since the derivative of the Lyapunov function is negative

outside a region, the system response cannot go outside of

this region. Indeed, the error is ultimately bounded.

Now, let e = zdz be the tracking error. It can be simplyfounded that e is the output of the following filter

= A + ( A+ 3(x,x ,u,)) (49)

e= (50)

The uncertain term 3(x,x,u,)) come from the obser-

vation error and the other terms that do not cancel by thecontrol signal. It is assumed that this term is bounded. Since

the filter (49) and (50) is stable and its input is bounded, the

tracking error, which is the output of the filter, is bounded.

In brief, in order to utilize the proposed control method-

ology, the following steps are required. First, The system

should be stated as (17) and (18). Then, an appropriate

diffeomorphism should be found using nonlinear control

theories to transform the system in the form of (22). Finally,

the observer and adaptation law are designed using (28), and

(38) and (39), respectively.

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(a) First state variable (p1 = x).

(b) Second state variable (p2 = ).

(c) Third state variable (p3 = ).

Fig. 2. Simulation results of the needle guidance problem with previouslyproposed output feedback linearization method of [12].

(a) First state variable (p1 = x).

(b) Second state variable (p2 = ).

(c) Third state variable (p3 = ).

Fig. 3. Simulation results of the needle guidance problem with the outputfeedback adaptive control method proposed in this paper.

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Fig. 4. The estimation of needle curvature ()

IV. SIMULATION R ESULTS

The proposed adaptive output feedback controller is em-

ployed to solve the problem of guiding a flexible bevel-tipneedle into a desired plane. The response of the system

is studied for two cases, first by applying the observer-

based controller proposed in [12] and second by applying the

proposed controller. In both cases the real needle curvature

is = 0.06 but our knowledge about this parameter is notexact. The known needle curvature is = 0.04. The initialconditions for the both cases are X0 = [0.1, 0.2, 0.8]

T.

In Fig. 2, the responses of the system with observer-

based controller [12] are shown. Since, with this controller

no parameter uncertainty is tolerated, the system response

should not be acceptable. Simulation results prove this

fact. In Fig. 3, the responses of the system with proposed

controller are plotted. This figure shows the good stabilityand convergence of the adaptive output feedback controller.

All state variables go near zero after a transient state. The

estimated needle curvature is shown in Fig. 4. Although this

parameter converges to a value, the converged value is not

the real one. This problem is not surprising, in that it can

be predicted theoretically from the given facts of Section

III. Since the error is ultimately bounded, we know that the

parameters just converge to a value. It is not any necessity

that the converged value be the real parameter.

V. CONCLUSIONS ANDF UTURE W ORKS

In this paper a novel observer-based controller is pro-posed for a class of nonlinear systems which are linearly

parametrized and feedback linearizable. The proposed strat-

egy is a modified version of a previously proposed adaptive

control scheme using high gain observer. The proposed

methodology is employed to guide a flexible bevel-tip needle

into a desired plane. A nonholonomic reduced order model

is considered for the needle which the needle curvature is

its only parameter. By utilizing the adaptive control, it is

possible to estimate the needle curvature in medical tasks

online. Through simulation results, it was demonstrated that

the proposed approach is quite effective for steering medical

needles into a desired plane.

Our next step is to evaluate the proposed methodology

with real data. One formidable barrier to achieve a good

performance in image guided tasks is measurement noise.

A more improved methodology can be proposed which

tolerates this barrier. The presented approach can also be

used with the automatic or manual path planning schemes.

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adaptive observer-based controller design for a class of nonlinear

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